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Mirrors > Home > MPE Home > Th. List > cnvrescnv | Structured version Visualization version GIF version |
Description: Two ways to express the corestriction of a class. (Contributed by BJ, 28-Dec-2023.) |
Ref | Expression |
---|---|
cnvrescnv | ⊢ ◡(◡𝑅 ↾ 𝐵) = (𝑅 ∩ (V × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-res 5567 | . . 3 ⊢ (◡𝑅 ↾ 𝐵) = (◡𝑅 ∩ (𝐵 × V)) | |
2 | 1 | cnveqi 5745 | . 2 ⊢ ◡(◡𝑅 ↾ 𝐵) = ◡(◡𝑅 ∩ (𝐵 × V)) |
3 | cnvin 6003 | . 2 ⊢ ◡(◡𝑅 ∩ (𝐵 × V)) = (◡◡𝑅 ∩ ◡(𝐵 × V)) | |
4 | cnvcnv 6049 | . . . 4 ⊢ ◡◡𝑅 = (𝑅 ∩ (V × V)) | |
5 | cnvxp 6014 | . . . 4 ⊢ ◡(𝐵 × V) = (V × 𝐵) | |
6 | 4, 5 | ineq12i 4187 | . . 3 ⊢ (◡◡𝑅 ∩ ◡(𝐵 × V)) = ((𝑅 ∩ (V × V)) ∩ (V × 𝐵)) |
7 | inass 4196 | . . 3 ⊢ ((𝑅 ∩ (V × V)) ∩ (V × 𝐵)) = (𝑅 ∩ ((V × V) ∩ (V × 𝐵))) | |
8 | inxp 5703 | . . . . 5 ⊢ ((V × V) ∩ (V × 𝐵)) = ((V ∩ V) × (V ∩ 𝐵)) | |
9 | inv1 4348 | . . . . . . 7 ⊢ (V ∩ V) = V | |
10 | 9 | eqcomi 2830 | . . . . . 6 ⊢ V = (V ∩ V) |
11 | ssv 3991 | . . . . . . . 8 ⊢ 𝐵 ⊆ V | |
12 | ssid 3989 | . . . . . . . 8 ⊢ 𝐵 ⊆ 𝐵 | |
13 | 11, 12 | ssini 4208 | . . . . . . 7 ⊢ 𝐵 ⊆ (V ∩ 𝐵) |
14 | inss2 4206 | . . . . . . 7 ⊢ (V ∩ 𝐵) ⊆ 𝐵 | |
15 | 13, 14 | eqssi 3983 | . . . . . 6 ⊢ 𝐵 = (V ∩ 𝐵) |
16 | 10, 15 | xpeq12i 5583 | . . . . 5 ⊢ (V × 𝐵) = ((V ∩ V) × (V ∩ 𝐵)) |
17 | 8, 16 | eqtr4i 2847 | . . . 4 ⊢ ((V × V) ∩ (V × 𝐵)) = (V × 𝐵) |
18 | 17 | ineq2i 4186 | . . 3 ⊢ (𝑅 ∩ ((V × V) ∩ (V × 𝐵))) = (𝑅 ∩ (V × 𝐵)) |
19 | 6, 7, 18 | 3eqtri 2848 | . 2 ⊢ (◡◡𝑅 ∩ ◡(𝐵 × V)) = (𝑅 ∩ (V × 𝐵)) |
20 | 2, 3, 19 | 3eqtri 2848 | 1 ⊢ ◡(◡𝑅 ↾ 𝐵) = (𝑅 ∩ (V × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 Vcvv 3494 ∩ cin 3935 × cxp 5553 ◡ccnv 5554 ↾ cres 5557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-res 5567 |
This theorem is referenced by: (None) |
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